Lecture 27 Sections 4.3 and 4.4
Section 4.3
Right Triangle Trigonometry
SOH CAH TOA
Cofunctions of complementary angles
Trigonometric identities
Section 4.4 Trigonometric Functions of Any
Angle
Reference angle
Angle of elevation and angle of depression
L27 - 1
Def. Let θ be an acute angle of a right triangle. Thesix trigonometric functions of the angle θ are definedas follows.
sin θ = csc θ =
cos θ = sec θ =
tan θ = cot θ =
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HYPOTENUSE OPPOSITE
10ADJACENT
AL1
OPP HYPSOHTpp
ADJ HYPCAHHYP ADJ
0 Toa ADIADJ OPP
SOH CAH TO Ap s
ex. If cos θ =3
4, sketch a right triangle with acute
angle θ, and find the other values of the five trigono-metric functions of θ.
Checkpoint: Lecture 27, problem 1
L27 - 3
COS 0 3 ADIdHYP ft HYP4 t
OPP PYTHAGOREANTHEOREM
42 32 t COPPIvj 0 16 9 t Copp I3g 7 PP5 Opp ItADJ
Opp 57
Sino ftp.pp E Csce HotpIp ftp.r4ff4coso
Iseco HffgI
z4tanooafj FI cot o Appt Ff Ffg 3
Theorem: Cofunctions of complementary anglesare equal. That is,
sin(90◦ − θ) = cos θ cos(90◦ − θ) = sin θ
tan(90◦ − θ) = cot θ cot(90◦ − θ) = tan θ
sec(90◦ − θ) = csc θ csc(90◦ − θ) = sec θ
a
b
c
θ
ex. Evaulate:
1) sin 36◦ − cos 54◦
2)tan 56◦
cot 34◦
Checkpoint: Lecture 27, problem 2
L27 - 4
EX COS 700 sin zoo icos 0 _ADI ibiBK sink200 sin 900700 Goo HYP if7051700 sin 900 o j OPP.I.be
HYP ICI
coso s.in 9oo o
90 54 36
sin 900 540 costs40 cos 540 cos 49 0
tan 900 340 Cot 340otL34 Cott 440
Trigonometric Identities
1. Reciprocal Identities
sin θ =1
csc θcsc θ =
1
sin θ
cos θ =1
sec θsec θ =
1
cos θ
tan θ =1
cot θcot θ =
1
tan θ
2. Quotient Identities
tan θ =sin θ
cos θcot θ =
cos θ
sin θ
3. Pythagorean Identities
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
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MUST KNOW THESE
a
L
sin 20 050 1
sin 20 050 1 ot l
atan 01 1 see 0
I t 050 ITino sin ltcotzo cs.co
ex. Let θ be an acute angle such that sin θ =1
3.
Use trigonometric identities to find
1) cos θ
2) tan θ
Checkpoint: Lecture 27, problem 3
L27 - 6
ACUTEANGLEsin 20 050 1 2 I ONLY
I cos 9 LPOSITIVEVALUES
f tcoso I LOSE too9cos't I tf FI3
COULD USE tan I see ZI3
OR QUICKER
tano siisfozfyj f.frEra FT
Lecture 27, Part II: Section 4.4
Trigonometric Functions of Any Angle
Def. Let θ be an angle in standard position and let(x, y) be a point on the terminal side. Ifr =
√
x2 + y2 is the distance from the origin to thepoint (x, y), then
sin θ = csc θ =
cos θ = sec θ =
tan θ = cot θ =
θ
(x,y)
NOTE: If x = 0, tan θ and sec θ are undefined. Ify = 0, cot θ and csc θ are undefined.
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NOT NECESSARILY A UNIT CIRCLE WHEREr I
y eF y
t er x
1 xx y
RADIUS HYPOTENUSEi rY on I
ex. Given point (−5, 12) on the terminal side of theangle θ, find the six trigonometric function values ofthe angle θ.
ex. If cos θ =3
5and θ lies in Quadrant IV, find the
values of all the trigonometric functions.
Checkpoint: Lecture 27, problem 4
L27 - 8
I 5,12f 2 Of 12
144 TOI 13 l
5Sino _r1 if cKO F
ELATESE tr
050 1 1 see 1 SOHCAHTOA Tr 13 TO1tano I L cot o yI
55 12
cos 0 1 3 Sec fr o
3Sino In csco Iy to 4
r i 4J i
tano f 4g Coto Ky ZITS
NEGATIVEB C QHI
Tty2 52
y2 25 9
y2 16
y 4
Signs of the Trigonometric Functions
ll ine
angent osine
A
T C
S
ex. Find the quadrant in which θ lies if tan θ < 0and sin θ < 0
ex. Given sec θ = 5 and sin θ < 0, find csc θ.
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ALL STUDENTSTAKE CALCULUS
1 it t t
l Ct
tano IY 15 NEGATIVE X IS POSITIVE
QI
1 Sec 0 55,2 I x L r 5
ALSO Y IS NEGATIVE SINCE sin 0 LO
WE ARE IN QII
O I
4 r 525 Hy Z
J 24 42Ys Tty I
rzy y 4 76co I off EE r 4 256
Reference Angles
Def. Let θ be an angle in standard position. Thereference angle is the acute angle θ′ formed bythe terminal side of θ and the x-axis.
θθ
θθ
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i iii i
q0 Oi n e0 0 01 01 1800
01 1800 O0 1 0 _IT0 11 0
w lo YO't Ii ri
0 11 0 01 0121101211 0O IT _O
OR 0 01 3600
0 180 to O 360 O
0 1800 0
ex. Find the reference angle for
1) θ =5π
3
2) θ = 870◦
Checkpoint: Lecture 27, problem 5
Evaluating Trigonometric Functions for Any
Angle
1. Find the reference angle θ′ associated with theangle θ.
2. Determine the sign of the trigonometric functionof θ by noting the quadrant in which θ lies.
3. The value of the trigonometric function of θ isthe same, except possibly for sign, as the value of thetrigonometric function of θ′.
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997O ZIT IT
31061 5115 O 61T3
3 IT3
O II3
FIND SMALLESTPOSITIVECOTERMINAL ANGLE THAT IS LESSTHAN360 1800 0 08700 3600 5100 y 01 180 O5100 3600 1500 O 01 1800 150001 300
ex. Use the reference angle to find
1) sin 240◦
2) cot 495◦
3) sin
(
16π
3
)
4) sec(
−π
4
)
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Sin2400 sin 600
o ioooi oi s.fi III240 1800101
01 600l i I
C it 4950 3600 1350
f35 Lot 4957 cot 13504501
cot 450
0 01 1800 EFI ray01 1800 0 1800 1350 450 JI I
2
6ft 217 1631 6,1 1031 Sin lbj sin 415
Ift 21T loft 6,1 431 Sin EsB
01 415 it 2It
4Iz LI27,1 817 7,1 secftfj seyttyj
seytfj fz fz.FIEiEzrz
O'IIIII oh.at E pti
ex. If tan θ =2
3and θ is in Quadrant III, find cos θ.
Method 1: Use trigonometric identities.
Method 2: Use the reference angle.
Checkpoint: Lecture 27, problem 6
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ta n O t I see20
Z t I see Zoe
TecateIg sect
secco i tf FI3
cos o seto3
tan D OPIADJ
3
iii so Affp fff 3
I 25 t l 35 h13 h Z h 13T
Applications
depressionangle of
elevationangle of
object
object
observer
observer
ex. A giant redwood tree casts a shadow 600 feetlong. Find the height of the tree if the angle ofelevation of the sun is 30◦.
L27 - 14
tan 300_Off 00TREE h
h 600 tan 300
1300 E IE t 4 60013K 600ftSHADOW h 20053 FT
fan 300 1IzIz Z
B
f E3TI33
ex. From a point on the ground 500 feet from thebase of a building, an observer finds that the angleof elevation to the top of the building is 30◦ and thatthe angle of elevation to the top of a flagpole atopthe building is 32◦. Find the height of the buildingand the length of the flagpole.
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If START WITH SMALLER b
BUILDING 0 IPod tan 300
500
BIGGER D
Opp h 500 BjHgtoh tan 320h 5O0B
f th 500 tan 320 3
f 500 tan 1320 hI STBSTITUTE h
f 5OOtanl3z 5O0z3